On the centralizer of the sum of commuting nilpotent elements. (English) Zbl 1106.20036
The main result of this paper answers a question of Pevtsova. The result is Theorem 21: Let \(X,Y\) be commuting nilpotent elements in \(M_n(k)\) and let \(A=X+tY\) with \(t\) an indeterminate. If \(k\) has characteristic \(p>0\), assume that \(A^{p-1}=0\). Then \(X,Y\in\text{Lie\,}R_uC\) where \(C\) is the centralizer of \(A\) in \(\text{GL}(n,k)\). Moreover, for an open subvariety of pairs \((a,b)\), \(X,Y\in\text{Lie\,}R_uC_{a,b}\), where \(C_{a,b}\) is the centralizer of \(aX+bY\) in \(\text{GL}(n,k)\). Some analogs are proved for other simple algebraic groups.
Reviewer: Robert Guralnick (Los Angeles)
MSC:
| 20G15 | Linear algebraic groups over arbitrary fields |
| 17B45 | Lie algebras of linear algebraic groups |
Keywords:
Lie algebras; commuting nilpotent elements; connected reductive algebraic groups; semisimple algebraic groupsReferences:
| [1] | Demazure, M.; Gabriel, P., Groupes Algébriques (1970), Masson: Masson North-Holland, Paris, Amsterdam · Zbl 0203.23401 |
| [2] | E. Friedlander, J. Pevtsova, \( \operatorname{\Pi;} \); E. Friedlander, J. Pevtsova, \( \operatorname{\Pi;} \) · Zbl 1128.20031 |
| [3] | Guralnick, R.; Liebeck, M.; MacPherson, D.; Seitz, G., Modules for algebraic groups with finitely many orbits on subspaces, J. Algebra, 196, 211-250 (1997) · Zbl 0906.20029 |
| [4] | Jantzen, J. C., Nilpotent orbits in representation theory, (Anker, J.-P.; Orsted, B., Lie Theory: Lie Algebras and Representations, Progress in Mathematics, vol. 228 (2004), Birkhäuser: Birkhäuser Boston), 1-211 · Zbl 1169.14319 |
| [5] | Kempf, G. R., Instability in invariant theory, Ann. Math., 108, 2, 299-316 (1978), MR 80c:20057 · Zbl 0406.14031 |
| [6] | Lang, S., Algebra (1993), Addison-Wesley: Addison-Wesley Readings, MA · Zbl 0848.13001 |
| [7] | Q. Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, Oxford University Press, Oxford, 2002 (translated from French by Reinie Erné).; Q. Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, Oxford University Press, Oxford, 2002 (translated from French by Reinie Erné). · Zbl 0996.14005 |
| [8] | McNinch, G. J., Nilpotent orbits over ground fields of good characteristic, Math. Annal., 329, 49-85 (2004), arXiv:math.RT/0209151 · Zbl 1141.17017 |
| [9] | G.J. McNinch, Optimal \(\operatorname{SL}_2\); G.J. McNinch, Optimal \(\operatorname{SL}_2\) · Zbl 1097.20040 |
| [10] | Premet, A., Nilpotent orbits in good characteristic and the Kempf-Rousseau theory, J. Algebra, 260, 338-366 (2003) · Zbl 1020.20031 |
| [11] | Premet, A., Nilpotent commuting varieties of reductive Lie algebras, Invent. Math., 154, 653-683 (2003) · Zbl 1068.17006 |
| [12] | Seitz, G. M., Unipotent elements, tilting modules, and saturation, Invent. Math., 141, 467-502 (2000) · Zbl 1053.20043 |
| [13] | T.A. Springer, Linear algebraic groups, second ed., Progress in Mathematics, vol. 9, Birkhäuser, Boston, 1998.; T.A. Springer, Linear algebraic groups, second ed., Progress in Mathematics, vol. 9, Birkhäuser, Boston, 1998. · Zbl 0927.20024 |
| [14] | T.A. Springer, R. Steinberg, Conjugacy classes, Seminar on Algebraic Groups and Related Finite Groups, The Institute for Advanced Study, Princeton, NJ, 1968/69, Lecture Notes in Mathematics, vol. 131, MR 42 #3091, Springer, Berlin, 1970, pp. 167-266.; T.A. Springer, R. Steinberg, Conjugacy classes, Seminar on Algebraic Groups and Related Finite Groups, The Institute for Advanced Study, Princeton, NJ, 1968/69, Lecture Notes in Mathematics, vol. 131, MR 42 #3091, Springer, Berlin, 1970, pp. 167-266. |
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